Problem: Simplify and expand the following expression: $ \dfrac{3}{2a + 18}+ \dfrac{5}{a + 7}+ \dfrac{4a}{a^2 + 16a + 63} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{3}{2a + 18} = \dfrac{3}{2(a + 9)}$ We can factor the quadratic in the third term: $ \dfrac{4a}{a^2 + 16a + 63} = \dfrac{4a}{(a + 9)(a + 7)}$ Now we have: $ \dfrac{3}{2(a + 9)}+ \dfrac{5}{a + 7}+ \dfrac{4a}{(a + 9)(a + 7)} $ The least common multiple of the denominators is: $ 2(a + 9)(a + 7)$ In order to get the first term over $2(a + 9)(a + 7)$ , multiply by $\dfrac{a + 7}{a + 7}$ $ \dfrac{3}{2(a + 9)} \times \dfrac{a + 7}{a + 7} = \dfrac{3(a + 7)}{2(a + 9)(a + 7)} $ In order to get the second term over $2(a + 9)(a + 7)$ , multiply by $\dfrac{2(a + 9)}{2(a + 9)}$ $ \dfrac{5}{a + 7} \times \dfrac{2(a + 9)}{2(a + 9)} = \dfrac{10(a + 9)}{2(a + 9)(a + 7)} $ In order to get the third term over $2(a + 9)(a + 7)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{4a}{(a + 9)(a + 7)} \times \dfrac{2}{2} = \dfrac{8a}{2(a + 9)(a + 7)} $ Now we have: $ \dfrac{3(a + 7)}{2(a + 9)(a + 7)} + \dfrac{10(a + 9)}{2(a + 9)(a + 7)} + \dfrac{8a}{2(a + 9)(a + 7)} $ $ = \dfrac{ 3(a + 7) + 10(a + 9) + 8a} {2(a + 9)(a + 7)} $ Expand: $ = \dfrac{3a + 21 + 10a + 90 + 8a}{2a^2 + 32a + 126} $ $ = \dfrac{21a + 111}{2a^2 + 32a + 126}$